That's a theory but I'm not convinced. As long as they have sympathiser here, like the catholic church, they will have someone here to provide shelter and money. It's more than they will get in Mexico.
At least they would be a much smaller drain on social services. It would also stop the "dumping" of labor that Mexico is engaging in.
If the Chinese were dumping cheap cars on the country everyone would be up in arms, but mexico can dump labor here and we don't even enact a tarriff.